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I have a problem as follows. Really appreciate if anyone can give me some suggestions.

I have $4000$ binary variables $\{x_0, x_1,...x_{3999}\}$ and $4000$ inequalities which have both binary addition (denoted as $+'$) and normal (arithmetic) addition (denoted as $+$).

For example, inequality $1$:

$$ (x_0 +' x_{39} +' x_{71} +' x_{3191}) + (x_{1} +' x_{44} +' x_{182} +' x_{2142}) + ... \leq 1 $$ inequality $2$: $$ (x_3 +' x_9 +' x_{39}) + (x_{1} +' x_{90}) + ... \leq 1 $$ ... inequality $4000$: $$ (x_{99} +' x_{51} +' x_{1191}) + (x_{171} +' x_{1441} +' x_{1821} +' x_{2142}) + ... <= 1 $$

Question 1: is there any way to tell if there is no solution rather than solving it?

Question 2: if there are some solutions, how to find some of them quickly?

Thank you very much, any suggestion is highly appreciated.

wjmccann
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user2883918
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1 Answers1

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It appears that your problem is a more complex version of the $3-SAT$ problem, which can be found and explained here.

As for your questions, it is unknown whether or not these types of problems can be solved efficiently (i.e. non-exponentially), however most theorize that you cannot. However if you were to find an algorithm that quickly solved the problem, there is quite the large bounty associated with it!

wjmccann
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  • thank you for your answer. I just googled and found that boolean operators can be replaced by adding some auxiliary variables with some linear constraints. (see here: https://cs.stackexchange.com/questions/12102/express-boolean-logic-operations-in-zero-one-integer-linear-programming-ilp). Then the problem can be solved by BIP. But the numbers of variables and constraints are increased significantly. Does anyone know any solver that can solve BIP with tens of thousands of variables and constraints? – user2883918 Jul 05 '18 at 04:59